Question

A rod (mass 3kg, length = 1 meters) is hanging vertically, hinged smoothly at its one end. A small ball of mass 1kg moving horizontally and perpendicularly to the rod with speed v, strikes the rod at its mid point and sticks to it. Find the minimum value of v (in m/sec) so that the rod completes its vertical circular motion subsequently.

Answer 1

$Conservationofangularmomentumaboutthehinge\frac{mVl}{2}=(\frac{M{l}^2}{3}+m{\left(\frac{l}{2}\right)}^2)\omega \Rightarrow \frac{V}{2}=\frac{5}{4}\omega ConservaationofMEfromthelowestpointtothehighest\frac{1}{2}(\frac{M{l}^2}{3}+m{\left(\frac{l}{2}\right)}^2){\omega }^2=(M+m)gl\Rightarrow \frac{1}{2}\times \frac{5}{4}{\omega }^2=4g\Rightarrow \frac{4}{25}{V}^2=\frac{32}{5}g\Rightarrow V=20$